Stochastic processes and randomness are vital features of mathematical modeling in biology.
Unfortunately analytical results are rarely available for even moderately complex
stochastic processes leaving simulation and numerical techniques the main avenues of attack.
We begin this work by exploring coupling bounds for birth-death processes, a fundamental
type of stochastic process that describes how populations of individuals change over
time. By forming a coupling between a truncated version of the process and the original
unbounded version, we are able to compute both moments and transition probabilities for
the true process within an acceptable error bound. Second, we present an algorithm design
framework for Interacting Particle Systems (IPSs). These are complex stochastic processes
with wide application to spatial phenomenon across many scientific disciplines. Here we describe
a method for efficiently sorting particles into classes based off of their type and spatial
configuration in such a fashion that reduces the spatial simulation to that of a non-spatial
well-mixed process, albeit with a more complicated update step. This also allows us to apply
a large suite of well-developed stochastic simulation algorithms to IPSs with little additional
coding cost. Third, we return to numerical methods, this time for multi-type branching
processes applied to gene therapy. We derive a series of ordinary differential equations that
govern the evolution of the probability generating function and provide a straightforward
numerical inversion approach to obtain marginalized probability distributions for probabilistic
quantities of interest. We provide examples of our techniques applied to lentiviral gene
therapy and the associated risk of oncogenesis in transplanted hematopoietic stem cell lines.
Finally, we conclude with a chapter on future directions, both related to the previous three
chapters as well as projects not previously addressed in this work.
